Question

A comet orbits the Sun and loses mass at a constant rate of 30,000kg/s. The only force acting on the comet is the gravitational force.

a) From the definition of torque in terms of angular momentum, torque=dL/dt, show that

dI/dt=-alphaI/omega (alpha= angular acceleration, omega=angular velocity)

b) The quantity omega/omega(with a dot on it), where (omega with a dot on it=domega/dt) is a natural time scale for the system's change in angular frequency (to see this, think about the units). Approximating the comet's orbit as a circle with constant radius, estimate the time scale for a comet of mass 10^14kg.

Answer 1

T = dL/dt: But L = I omega = > dL/dt = I d omega/dt + omega dI/dt T = I d omega/dt + omega dI/dt = I alpha + omega dI/dt Here Net torque should be zero I alpha + omega dI/dt = 0 = > dI/dt = -I alpha/ omega Given dm/dt = 3 times 10^4 kg/s: m = 10^14 kg: r = R & r = 0 from (1) omega = (-1/I alpha dI/dt)^-1 omega omega = alpha omega/omega = omega/alpha = (-I/dI/dt) omega/omega = -I/(dI/dt) = T But I = mR^2: T = times scale dI/dt + I/T = 0 dI/dt R^2 dm/dt = R^2 times 3 times 10^4 3 times 10^4 R^2 + mR^2 /T = 0 = > T = -10^14 /3 times 10^4 = -3.33 times 10^9